Proof of The Observation-Centred Fluctuation at an Intermediate Time: Restriction of the Mean-Field Flow and of the Realized Control to a Shorter Horizon, Representation through the Aggregate State and the Record Prefix, and Transport of Its Joint Law with the Record to the Synthetic Copy

Throughout, the composition $g\circ f$ of a map $f$ measurable with respect to $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ and a map $g$ measurable with respect to $\mathcal{F}_{2}$ and $\mathcal{F}_{3}$ is measurable with respect to $\mathcal{F}_{1}$ and $\mathcal{F}_{3}$, since $(g\circ f)^{-1}(E)=f^{-1}(g^{-1}(E))$ for every set $E$ and measurability is the requirement that preimages of measurable sets be measurable; we use this without further comment. For a map $f$ on $[0,T]$ we write $f|_{[0,s]}$ for its restriction to $[0,s]$.

Claim 1. The Data paragraph of The Realized Control as the Record-Frozen Control at the Observation Record, and Measurability of the Path-and-Record Closeness Set with horizon $s$ requires: the natural numbers $N\ge1$, $l\ge2$, $\tilde{l}\ge1$, $m\ge1$; a real number $\Lambda\ge0$ and an affine-controlled transition-rate family $(\beta_{0},\beta_{1})$ with Lipschitz constant $\Lambda$ whose control set $\mathcal{A}$ is nonempty, convex and compact; its transition-rate family $\beta$ with the control bound $R^{\mathcal{A}}$ and the point $a_{0}\in\mathcal{A}$; an observation-rate family on $l$ states with $\tilde{l}$ channels; a real horizon (here $s>0$); an $N$-agent driving system; an $\mathcal{A}$-valued observation-driven control policy with horizon $s$, control dimension $m$ and $\tilde{l}$ channels; a solution of the controlled $N$-agent dynamics on $[0,s]$ for these data; and, for the setting of The Realized Control of the Controlled N-Agent Dynamics as a Random Element of the Control Set adopted there, a sequence in $L^{2}([0,s];\mathbb{R}^{m})$ whose set of terms is dense, from which its metric is formed. All of these are supplied by the adopted data with the horizon $s$: the numbers, the affine family, $\mathcal{A}$, $\beta$, $R^{\mathcal{A}}$, $a_{0}$ and $\tilde{\beta}$ are those adopted and do not involve the horizon; the driving system is the adopted one, the same driving system serving for the horizon $s$ by claim 2 of Restriction of a Solution of the Controlled N-Agent Dynamics to a Shorter Horizon: the Truncated Policy, the Restricted Solution, Its Filtrations, and Its Record as the Prefix of the Record; the policy is $h^{(s)}$, which is an $\mathcal{A}$-valued observation-driven control policy with horizon $s$, control dimension $m$ and $\tilde{l}$ channels by claim 1 of that lemma when $s<T$ and because $h^{(T)}=h$ when $s=T$; the solution is the restricted solution, which is a solution on $[0,s]$ for $h^{(s)}$ and the same driving system, rate families and control set by claim 2 of that lemma when $s<T$ and by definition when $s=T$; and the dense sequence is the one fixed in the statement; such a sequence exists, because the dense subset of $L^{2}([0,s];\mathbb{R}^{m})$ furnished by Separability of the Lebesgue Space of Square-Integrable Vector-Valued Functions with horizon $s$ is countable and nonempty (it contains the class of the zero map), hence is the set of terms of a sequence by Countable Set. The comparison data and the path data of that lemma are supplied by $S^{*}_{t}=0_{\mathbb{R}^{l}}$ ($t\in[0,s]$), whose components are constant, hence continuous, with $|S^{*}_{t}|=0\le K^{*}=0$; by $A_{t}=a_{0}$ ($t\in[0,s]$), a map into $\mathcal{A}$ with constant, hence $\mathcal{B}_{[0,s]}$-measurable, components (claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions); and by the nonempty finite set $E=\{0_{\mathbb{R}^{l}}\}$. Thus the whole setting of that lemma with horizon $s$ is instantiated, and its claim 1 is available for the restricted solution. The realized-control lemma with horizon $s$ requires only these data together with the convexity and compactness of $\mathcal{A}$ and a bound on $\mathcal{A}$, for which $R^{\mathcal{A}}$ serves, as noted in The Realized Control as the Record-Frozen Control at the Observation Record, and Measurability of the Path-and-Record Closeness Set. The flow stability lemma Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls requires an affine-controlled transition-rate family and a real horizon, here $(\beta_{0},\beta_{1})$ and $s>0$. The record-frozen control lemma The Record-Frozen Control as a Measurable Map into the Weakly Metrized Control Set, and the Record-Frozen Mean-Field Flow: Joint Measurability, Flow Equation, and Measurability in the Record requires the setting of the flow stability lemma (just verified), a natural number $\tilde{l}\ge1$, an $\mathcal{A}$-valued observation-driven control policy with horizon $s$, control dimension $m$ and $\tilde{l}$ channels (here $h^{(s)}$), a point of $\Delta^{l}$ in the role of $x_{0}$ (here $z_{0}$), and a sequence in $L^{2}([0,s];\mathbb{R}^{m})$ with dense set of terms from which its metric $\rho$ is formed (the fixed sequence). Finally, The Observation Filtration of a Solution of the Controlled N-Agent Dynamics is Generated, up to Null Sets, by the Observation Record up to that Time requires the setting of the controlled $N$-agent dynamics with $\mathcal{A}$ convex and compact, the rate families with their rate bounds, the horizon $T$, the driving system, the $\mathcal{A}$-valued policy $h$, the given solution, a real number $s$ with $0<s\le T$, and reconstruction data for the driving system and $h^{(s)}$ with horizon $s$ (and, when $s=T$, for $h$ with horizon $T$, which is the same requirement since $h^{(T)}=h$); all are adopted, the reconstruction data being those fixed in the statement, and that lemma forms the truncated policy, the restricted solution and $W^{(s)}$ exactly as the statement does.

Consequently: $\hat{\alpha}^{(s)}$ is the map of claim 2 of the realized-control lemma with horizon $s$, defined from a family furnished by its claim 1; the paths $a^{(s),r}$ are defined by The Record-Frozen Control Path and Record-Frozen Policy for the policy $h^{(s)}$; $\mathcal{U}^{(s)}_{\mathcal{A}}$ and $\mathsf{S}^{(s)}$ are furnished by claim 2 of the flow stability lemma with horizon $s$; each $a^{(s),r}$ represents an element of $\mathcal{U}^{(s)}_{\mathcal{A}}$ by claim 1 of the record-frozen control lemma, so that $\Phi^{(s),r}_{t}=\mathsf{S}^{(s)}_{t}(z_{0},a^{(s),r})$ is the record-frozen flow of its claim 4 with base point $z_{0}$; $\hat{\alpha}^{(s)}(\omega)\in\mathcal{U}^{(s)}_{\mathcal{A}}$ for every $\omega$ by claim 3 of the realized-control lemma with horizon $s$, so that $\Phi^{(s)}_{t}(\omega)$ is defined; the map $r\mapsto\Phi^{(s),r,\gamma}_{t}$ is $\mathcal{R}_{s}$-measurable by claim 5 of the record-frozen control lemma; and $W^{(s)}$ is measurable with respect to $\mathcal{F}$ and $\mathcal{R}_{s}$ by claim 1 of The Observation Filtration of a Solution of the Controlled N-Agent Dynamics is Generated, up to Null Sets, by the Observation Record up to that Time.

Claim 2. Measurability of the restriction. Let $\gamma\in\{1,\dots,m\}$ and let $E$ be a Borel subset of the real line. Since $u^{\gamma}$ is measurable with respect to $\mathcal{B}_{[0,T]}$, the set $(u^{\gamma})^{-1}(E)$ belongs to $\mathcal{B}_{[0,T]}$, so by the definition of $\mathcal{B}_{[0,T]}$ in Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval it equals $S\cap[0,T]$ for some Borel set $S\subseteq\mathbb{R}$. Then $(u^{\gamma}|_{[0,s]})^{-1}(E)=(u^{\gamma})^{-1}(E)\cap[0,s]=S\cap[0,s]$, which belongs to $\mathcal{B}_{[0,s]}$ by the same definition. Hence the components of $u|_{[0,s]}$ are $\mathcal{B}_{[0,s]}$-measurable, and $u|_{[0,s]}$ takes its values in $\mathcal{A}$; so claim 1 of Existence and Uniqueness of the Generalized Mean-Field Trajectory for a Measurable Control applies to the horizon $s$, the initial value $x$ and the control $u|_{[0,s]}$, and $S^{(s),u|_{[0,s]}}$ is defined.

The restriction of $S^{u}$ solves the horizon-$s$ equation. Write $y=S^{u}|_{[0,s]}$. By claim 1 of the existence theorem, $(S^{u},u)$ is a generalized mean-field trajectory pair for $(\beta_{0},\beta_{1})$ with horizon $T$, with $S^{u}_{v}\in\Delta^{l}$ for every $v$ and $S^{u}_{0}=x$. By condition 1 of that definition each component $S^{u,\gamma}$ is continuous on $[0,T]$, the interval and the real line carrying the metric of the real line; hence each $y^{\gamma}$ is continuous on $[0,s]$: given $t_{0}\in[0,s]$ and $\varepsilon>0$, a $\delta>0$ with $|S^{u,\gamma}_{t}-S^{u,\gamma}_{t_{0}}|<\varepsilon$ for all $t\in[0,T]$ with $|t-t_{0}|<\delta$ serves in particular for all such $t\in[0,s]$, which is the requirement of continuity of $y^{\gamma}$ at $t_{0}$ relative to $[0,s]$. For every $\gamma\in\{1,\dots,l\}$, condition 2 of that definition gives, with $f(v)=b^{\gamma}(S^{u}_{v},u_{v})$ for $v\in[0,T]$,

$$S^{u,\gamma}_{t}=x^{\gamma}+\int_{[0,t]}f|_{[0,t]}\,d\lambda_{[0,t]}\qquad(0<t\le T),\qquad S^{u,\gamma}_{0}=x^{\gamma},$$

the integral being the Lebesgue integral over the compact interval $[0,t]$; as recorded in that definition, $f$ is measurable with respect to $\mathcal{B}_{[0,T]}$ and bounded in absolute value by $K_{b}=2\sqrt{l}(l-1)B$ (the Euclidean norm bound $|b|\le K_{b}$ of claim 4 of The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data with the coordinate bound of claim 4 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square). Fix $t\in(0,s]$ and $\gamma$. The function $f\mathbf{1}_{[0,t]}$ on $[0,T]$ is $\mathcal{B}_{[0,T]}$-measurable (claims 1 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, $[0,t]\in\mathcal{B}_{[0,T]}$) and integrable with respect to $\lambda_{[0,T]}$: $|f\mathbf{1}_{[0,t]}|\le K_{b}$, the constant $K_{b}$ has the finite integral $K_{b}\lambda_{[0,T]}([0,T])=K_{b}T$ (The Integral of an Indicator Function is the Measure of the Set, positive homogeneity in Linearity and Monotonicity of the Lebesgue Integral, claim 1 of the toolkit), so $|f\mathbf{1}_{[0,t]}|$ has finite integral by monotonicity of the integral, which is integrability. Let $g:\mathbb{R}\to\mathbb{R}$ be the function equal to $f$ on $[0,t]$ and to $0$ elsewhere. Then $g$ is the zero extension of each of the three functions $f\mathbf{1}_{[0,t]}$ on $[0,T]$, $f|_{[0,s]}\mathbf{1}_{[0,t]}$ on $[0,s]$ and $f|_{[0,t]}$ on $[0,t]$, because $[0,t]\subseteq[0,s]\subseteq[0,T]$. By claim 2 of the toolkit applied on $[0,T]$, $g$ is Borel measurable and integrable with respect to Lebesgue measure $\lambda$, with $\int_{[0,T]}f\mathbf{1}_{[0,t]}\,d\lambda_{[0,T]}=\int_{\mathbb{R}}g\,d\lambda$; by the same claim applied on $[0,s]$ and on $[0,t]$ in the converse direction, $f|_{[0,s]}\mathbf{1}_{[0,t]}$ is $\mathcal{B}_{[0,s]}$-measurable and integrable with respect to $\lambda_{[0,s]}$, $f|_{[0,t]}$ is $\mathcal{B}_{[0,t]}$-measurable and integrable with respect to $\lambda_{[0,t]}$, and

$$\int_{[0,s]}f|_{[0,s]}\mathbf{1}_{[0,t]}\,d\lambda_{[0,s]}=\int_{\mathbb{R}}g\,d\lambda=\int_{[0,t]}f|_{[0,t]}\,d\lambda_{[0,t]}=\int_{[0,T]}f\mathbf{1}_{[0,t]}\,d\lambda_{[0,T]} .$$

(This chain shows that the three readings of an integral over $[0,t]$ --- as an integral of the restriction with respect to $\lambda_{[0,t]}$, or of the integrand multiplied by the indicator of $[0,t]$ with respect to $\lambda_{[0,s]}$ or to $\lambda_{[0,T]}$ --- agree, so the identity below does not depend on which reading of condition 2 is adopted.) Since $f(v)=b^{\gamma}(y_{v},u|_{[0,s]}(v))$ for $v\in[0,s]$, the restriction to $[0,t]$ of the map $v\mapsto b^{\gamma}(y_{v},u|_{[0,s]}(v))$ on $[0,s]$ is $f|_{[0,t]}$, and the display for $S^{u,\gamma}_{t}$ shows

$$y^{\gamma}_{t}=x^{\gamma}+\int_{[0,t]}\Bigl(v\mapsto b^{\gamma}\bigl(y_{v},u|_{[0,s]}(v)\bigr)\Bigr)\Big|_{[0,t]}\,d\lambda_{[0,t]}\qquad\text{for all }t\in(0,s],\ \gamma\in\{1,\dots,l\},$$

the right-hand side being the Lebesgue integral over the compact interval $[0,t]$ of the integrand required in condition 2 of the definition of a generalized mean-field trajectory pair with horizon $s$ (for $t=0$, where that condition reads $y^{\gamma}_{0}=x^{\gamma}$ by its convention that the integral is $0$, it holds because $y_{0}=S^{u}_{0}=x$). Together with the continuity of the components of $y$, its values in $\Delta^{l}$, and the $\mathcal{B}_{[0,s]}$-measurability and $\mathcal{A}$-valuedness of $u|_{[0,s]}$ (first paragraph), this shows that $(y,u|_{[0,s]})$ is a generalized mean-field trajectory pair for $(\beta_{0},\beta_{1})$ with horizon $s$ whose value at $t=0$ is $x$. By claim 2 (uniqueness) of the existence theorem with horizon $s$, applied to the control $u|_{[0,s]}$ and the initial value $x$, $y=S^{(s),u|_{[0,s]}}$; that is, $S^{(s),u|_{[0,s]}}_{t}=S^{u}_{t}$ for every $t\in[0,s]$.

The class of the restriction. Let $\xi\in\mathcal{U}_{\mathcal{A}}$ and let $u$ be an admissible representative of $\xi$: by claim 2 of the flow stability lemma, $u$ is a representative of $\xi$ (hence a square-integrable map $[0,T]\to\mathbb{R}^{m}$ in the sense of claim 1 of The Lebesgue Space of Square-Integrable Vector-Valued Functions on a Compact Interval, whose components are therefore $\mathcal{B}_{[0,T]}$-measurable) with $u(t)\in\mathcal{A}$ for every $t\in[0,T]$. By the first paragraph the components of $u|_{[0,s]}$ are $\mathcal{B}_{[0,s]}$-measurable. Write $u_{s}=u|_{[0,s]}$. Then $|u_{s}(v)|\le R^{\mathcal{A}}$ for every $v\in[0,s]$, so the nonnegative $\mathcal{B}_{[0,s]}$-measurable function $|u_{s}|^{2}$ (measurable by claims 3 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, being the sum of the squares of the components) is bounded by the constant $(R^{\mathcal{A}})^{2}$, whose integral with respect to $\lambda_{[0,s]}$ is $(R^{\mathcal{A}})^{2}\lambda_{[0,s]}([0,s])=(R^{\mathcal{A}})^{2}s$ by The Integral of an Indicator Function is the Measure of the Set, positive homogeneity of the integral (Linearity and Monotonicity of the Lebesgue Integral) and claim 1 of the toolkit; by monotonicity of the integral $\int_{[0,s]}|u_{s}|^{2}\,d\lambda_{[0,s]}\le(R^{\mathcal{A}})^{2}s<\infty$. Hence $u_{s}=u|_{[0,s]}$ is square-integrable on $[0,s]$; its values lie in $\mathcal{A}$ at every point, so its class $[u|_{[0,s]}]$ belongs to $\mathcal{U}^{(s)}_{\mathcal{A}}$ by The Set of Controls with Values in a Prescribed Subset of Euclidean Space (with the empty null set), and $u|_{[0,s]}$ is an admissible representative of it. By claim 2 of the flow stability lemma with horizon $s$, $\mathsf{S}^{(s)}(x,[u|_{[0,s]}])$ is the map $S^{(s),u|_{[0,s]}}$, and with horizon $T$, $\mathsf{S}(x,\xi)$ is the map $S^{u}$; the identity $\mathsf{S}^{(s)}_{t}(x,[u|_{[0,s]}])=\mathsf{S}_{t}(x,\xi)$ for $t\in[0,s]$ is therefore the identity just proved.

Claim 3. The realized controls. Let $\omega\in\Omega$ and $t\in[0,s]$. If $\omega\notin\Omega_{0}$, then $\hat{\alpha}(t,\omega)=a_{0}=\hat{\alpha}^{(s)}(t,\omega)$ by the definition in claim 2 of the realized-control lemma at both horizons, the regular event of the restricted solution being $\Omega_{0}$ (claim 2 of Restriction of a Solution of the Controlled N-Agent Dynamics to a Shorter Horizon: the Truncated Policy, the Restricted Solution, Its Filtrations, and Its Record as the Prefix of the Record, or by definition when $s=T$). If $\omega\in\Omega_{0}$, then by claim 2 of the realized-control lemma $\hat{\alpha}(t,\omega)=\alpha_{t}(\omega)$, the value at $(t,\omega)$ of the control process of the given solution, and $\hat{\alpha}^{(s)}(t,\omega)=\alpha_{t}(\omega)$, the value of the control process of the restricted solution, which is $(\alpha_{t})_{t\in[0,s]}$. Hence $\hat{\alpha}^{(s)}(t,\omega)=\hat{\alpha}(t,\omega)$ for all $\omega\in\Omega$ and $t\in[0,s]$; that is, the path $\hat{\alpha}^{(s)}(\omega)$ is the restriction $\hat{\alpha}(\omega)|_{[0,s]}$.

The realized flows. Fix $\omega\in\Omega$. By claim 3 of the realized-control lemma the path $\hat{\alpha}(\omega)$ is an admissible representative of the element $\hat{\alpha}(\omega)\in\mathcal{U}_{\mathcal{A}}$, and $\Phi_{t}(\omega)=\mathsf{S}_{t}(z_{0},\hat{\alpha}(\omega))$. By claim 2 applied to $x=z_{0}$, $\xi=\hat{\alpha}(\omega)$ and this representative, $\mathsf{S}^{(s)}_{t}(z_{0},[\hat{\alpha}(\omega)|_{[0,s]}])=\Phi_{t}(\omega)$ for $t\in[0,s]$. By the previous paragraph $\hat{\alpha}(\omega)|_{[0,s]}$ is the path $\hat{\alpha}^{(s)}(\omega)$, whose class is the element $\hat{\alpha}^{(s)}(\omega)$ of $\mathcal{U}^{(s)}_{\mathcal{A}}$ (claim 3 of the realized-control lemma with horizon $s$); hence $\Phi^{(s)}_{t}(\omega)=\mathsf{S}^{(s)}_{t}(z_{0},\hat{\alpha}^{(s)}(\omega))=\Phi_{t}(\omega)$ for every $t\in[0,s]$.

The record prefix. Let $\omega\in\Omega_{0}$ and $t\in[0,s]$. Claim 1 of The Realized Control as the Record-Frozen Control at the Observation Record, and Measurability of the Path-and-Record Closeness Set, applied with horizon $s$ (its hypotheses hold by claim 1), to the restricted solution, whose observation record is $W^{(s)}$, gives $\hat{\alpha}^{(s)}(t,\omega)=a^{(s),W^{(s)}(\omega)}(t)$. Since this holds for every $t\in[0,s]$, the path $\hat{\alpha}^{(s)}(\omega)$ and the path $a^{(s),W^{(s)}(\omega)}$ coincide, so they represent the same element of $L^{2}([0,s];\mathbb{R}^{m})$; therefore $\Phi^{(s)}_{t}(\omega)=\mathsf{S}^{(s)}_{t}(z_{0},a^{(s),W^{(s)}(\omega)})=\Phi^{(s),W^{(s)}(\omega)}_{t}$, and by the previous paragraph $\Phi_{t}(\omega)=\Phi^{(s),W^{(s)}(\omega)}_{t}$.

Claim 4. Values in the lattice, and the diameter of the simplex. By the derived notation of Solution of the Controlled N-Agent Dynamics, $\Sigma^{\gamma}_{t}(\omega)=\frac1N\sum_{i=1}^{N}\eta^{i,\gamma}_{t}(\omega)$ with each $\eta^{i,\gamma}_{t}(\omega)\in\{0,1\}$, and $\Sigma_{t}(\omega)\in\Delta^{l}$; thus $N\Sigma^{\gamma}_{t}(\omega)$ is the number of agents $i$ with $\sigma^{i}_{t}(\omega)=\gamma$, which is $0$ or a natural number, so $\Sigma_{t}(\omega)\in\mathbb{G}_{N}$ by the definition of the aggregate lattice, for every $\omega\in\Omega$ and $t\in[0,T]$. For $x\in\Delta^{l}$ one has $x^{\gamma}\ge0$ and $\sum_{\gamma}x^{\gamma}=1$, so $|x|^{2}=\sum_{\gamma}(x^{\gamma})^{2}\le\bigl(\sum_{\gamma}x^{\gamma}\bigr)^{2}=1$ (claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square; the inequality because the square of a sum of nonnegative numbers is the sum of their squares plus the nonnegative cross terms), hence $|x|\le1$, and for $x,y\in\Delta^{l}$ the triangle inequality (claim 6 there, applied to $x-y=x+(-y)$, with $|-y|=|y|$ by claim 5) gives $|x-y|\le|x|+|y|\le2$.

Measurability and bound for $\Psi_{t}$. Let $t\in[0,s]$ and $\gamma\in\{1,\dots,l\}$. The map $(x,r)\mapsto x^{\gamma}$ on $\mathbb{G}_{N}\times\mathbf{R}_{s}$ is the composition of the coordinate projection $(x,r)\mapsto x$, measurable with respect to $\mathcal{P}(\mathbb{G}_{N})\otimes\mathcal{R}_{s}$ and $\mathcal{P}(\mathbb{G}_{N})$ by claim 5 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable, with the map $x\mapsto x^{\gamma}$ on $\mathbb{G}_{N}$, which is measurable with respect to $\mathcal{P}(\mathbb{G}_{N})$ because every subset of $\mathbb{G}_{N}$ belongs to $\mathcal{P}(\mathbb{G}_{N})$. The map $(x,r)\mapsto\Phi^{(s),r,\gamma}_{t}$ is the composition of the projection $(x,r)\mapsto r$ (measurable by the same claim) with $r\mapsto\Phi^{(s),r,\gamma}_{t}$, which is $\mathcal{R}_{s}$-measurable by claim 1. Hence $\Psi^{\gamma}_{t}(x,r)=\sqrt{N}\,x^{\gamma}-\sqrt{N}\,\Phi^{(s),r,\gamma}_{t}$ is measurable by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. Since $x\in\mathbb{G}_{N}\subseteq\Delta^{l}$ and $\Phi^{(s),r}_{t}\in\Delta^{l}$, the first paragraph and claim 5 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square give $|\Psi_{t}(x,r)|=\sqrt{N}\,|x-\Phi^{(s),r}_{t}|\le2\sqrt{N}$.

The representation. For $\omega\in\Omega_{0}$ and $t\in[0,s]$, claim 3 gives $\Phi_{t}(\omega)=\Phi^{(s),W^{(s)}(\omega)}_{t}$, so $X'_{t}(\omega)=\sqrt{N}(\Sigma_{t}(\omega)-\Phi^{(s),W^{(s)}(\omega)}_{t})=\Psi_{t}(\Sigma_{t}(\omega),W^{(s)}(\omega))$, where $\Sigma_{t}(\omega)\in\mathbb{G}_{N}$ by the first paragraph.

Random variables and square-integrability. Let $t\in[0,s]$. Each $\Sigma^{\gamma}_{t}=\frac1N\sum_{i}\eta^{i,\gamma}_{t}$ is a random variable: $\eta^{i,\gamma}_{t}$ is the indicator of the event $\{\sigma^{i}_{t}=\gamma\}$, which belongs to $\mathcal{F}$ because $\sigma^{i}_{t}$ is a random variable with values in $\{1,\dots,l\}$ and $\{\gamma\}$ is a Borel set, and indicators and linear combinations of random variables are random variables by claims 1 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. For $\Phi^{\gamma}_{t}$, by claim 3, $\Phi^{\gamma}_{t}=\mathbf{1}_{\Omega_{0}}\,(\Phi^{(s),W^{(s)},\gamma}_{t})+\mathbf{1}_{\Omega\setminus\Omega_{0}}\,\mathsf{S}^{\gamma}_{t}(z_{0},\bar{a}_{0})$, where $\Phi^{(s),W^{(s)},\gamma}_{t}$ denotes the composition $\omega\mapsto\Phi^{(s),W^{(s)}(\omega),\gamma}_{t}$, measurable as the composition of $W^{(s)}$ (measurable by claim 1) with the $\mathcal{R}_{s}$-measurable map $r\mapsto\Phi^{(s),r,\gamma}_{t}$, and $\bar{a}_{0}$ denotes the constant path with value $a_{0}$ (an admissible representative of an element of $\mathcal{U}_{\mathcal{A}}$, being constant hence measurable and bounded, and $\mathcal{A}$-valued), which is the path $\hat{\alpha}(\omega)$ for every $\omega\notin\Omega_{0}$; the displayed expression is a random variable by claims 1, 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, $\Omega_{0}$ being an event. Hence each $X'^{\gamma}_{t}=\sqrt{N}\Sigma^{\gamma}_{t}-\sqrt{N}\Phi^{\gamma}_{t}$ is a random variable (claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions). Since $\Sigma_{t}(\omega)$ and $\Phi_{t}(\omega)$ lie in $\Delta^{l}$ for every $\omega$ and every $t\in[0,T]$, the first paragraph gives $|X'_{t}(\omega)|\le2\sqrt{N}$ for all $\omega\in\Omega$ and $t\in[0,T]$; then, for $t\in[0,s]$, $|X'^{\gamma}_{t}|\le|X'_{t}|\le2\sqrt{N}$ (claim 4 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square) and, for $\mathbf{c}\in\mathbb{R}^{l}$, $|\mathbf{c}\cdot X'_{t}|\le|\mathbf{c}|\,|X'_{t}|\le2\sqrt{N}|\mathbf{c}|$ by The Cauchy-Schwarz Inequality on $\mathbb{R}^n$, and $\mathbf{c}\cdot X'_{t}$ is a random variable by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. A random variable $Z$ bounded by a constant $M$ is square-integrable: $Z^{2}\le M^{2}$ everywhere and the integral of the constant $M^{2}$ with respect to the probability measure $P$ is $M^{2}P(\Omega)=M^{2}$ (The Integral of an Indicator Function is the Measure of the Set and positive homogeneity in Linearity and Monotonicity of the Lebesgue Integral), so $\mathbb{E}[Z^{2}]\le M^{2}<\infty$ by monotonicity of the integral. This gives the square-integrability of each $X'^{\gamma}_{t}$ and of $\mathbf{c}\cdot X'_{t}$.

Joint measurability with the record prefix. Let $t\in[0,s]$ and $\mathbf{c}\in\mathbb{R}^{l}$. The product $\sigma$-algebra $\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$ is generated by the rectangles $E\times A$ with $E\in\mathcal{B}(\mathbb{R})$ and $A\in\mathcal{R}_{s}$ (Product Sigma-Algebra), and the preimage of such a rectangle under $\omega\mapsto(\mathbf{c}\cdot X'_{t}(\omega),W^{(s)}(\omega))$ is $(\mathbf{c}\cdot X'_{t})^{-1}(E)\cap(W^{(s)})^{-1}(A)\in\mathcal{F}$, both sets belonging to $\mathcal{F}$ by the preceding paragraphs and claim 1. By claim 2 of Generator Criterion for Measurability the map is measurable with respect to $\mathcal{F}$ and $\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$.

Claim 5. Preliminaries. Under the additional assumptions, the setting of Law Identity between the N-Agent Aggregate Path with Its Observation Record and the Synthetic Copy's Regularised Path with Its Record is instantiated with horizon $s$; its objects are the synthetic copy with $\Omega^{\sharp}=\Omega^{\flat}\times\mathbb{R}^{d}\times\mathbf{R}_{s}$, its record $\mathsf{D}$, the path space $\mathsf{Path}=\mathsf{Path}(\mathbb{G}_{N},s)$ with its $\sigma$-algebra $\mathcal{C}$ generated by the sets $\{p:p(v)=y\}$ ($v\in[0,s]$, $y\in\mathbb{G}_{N}$), the maps $\Pi$ on $\Omega$ and $\Pi^{\sharp}$ on $\Omega^{\sharp}$ with $\Pi^{\sharp}(\omega^{\flat},\theta,r)(v)=\bar{\Sigma}^{\sharp,r}_{v}(\omega^{\flat})$, and the set $D_{0}=\{\Sigma_{0}=x_{0}\}$. By claim 1 of that lemma, $D_{0}\in\mathcal{F}$ (so that $P(D_{0})=1$ by assumption), $\Pi^{\sharp}$ takes its values in $\mathsf{Path}$ and is measurable with respect to $\mathcal{F}^{\sharp}$ and $\mathcal{C}$, and $\mathsf{D}$ is measurable with respect to $\mathcal{F}^{\sharp}$ and $\mathcal{R}_{s}$. Since $\bar{\Sigma}^{\sharp,\mathsf{D}}_{s}(\omega^{\flat},\theta,r)=\bar{\Sigma}^{\sharp,r}_{s}(\omega^{\flat})=\Pi^{\sharp}(\omega^{\flat},\theta,r)(s)$ and every path in $\mathsf{Path}$ takes its values in $\mathbb{G}_{N}$ (The Space of Piecewise Constant Paths in a Finite Set: Measurability of Evaluations, Restrictions, Left Limits, and Occupation Integrals), $\bar{\Sigma}^{\sharp,\mathsf{D}}_{s}$ takes its values in $\mathbb{G}_{N}$; and for every $y\in\mathbb{G}_{N}$ the set $\{\bar{\Sigma}^{\sharp,\mathsf{D}}_{s}=y\}=(\Pi^{\sharp})^{-1}(\{p:p(s)=y\})$ belongs to $\mathcal{F}^{\sharp}$, so that $\{\bar{\Sigma}^{\sharp,\mathsf{D}}_{s}\in A\}=\bigcup_{y\in A}\{\bar{\Sigma}^{\sharp,\mathsf{D}}_{s}=y\}\in\mathcal{F}^{\sharp}$ for every $A\subseteq\mathbb{G}_{N}$, the union being finite because $\mathbb{G}_{N}\subseteq\{x\in\mathbb{R}^{l}:Nx^{\gamma}\in\{0,1,\dots,N\}\text{ for all }\gamma\}$ is finite (each $Nx^{\gamma}$ is a nonnegative integer at most $N$ because $0\le x^{\gamma}\le1$ on $\Delta^{l}$). Thus $\bar{\Sigma}^{\sharp,\mathsf{D}}_{s}$ is measurable with respect to $\mathcal{F}^{\sharp}$ and $\mathcal{P}(\mathbb{G}_{N})$, and the pair $(\bar{\Sigma}^{\sharp,\mathsf{D}}_{s},\mathsf{D})$ is measurable with respect to $\mathcal{F}^{\sharp}$ and $\mathcal{P}(\mathbb{G}_{N})\otimes\mathcal{R}_{s}$ by the rectangle argument of the last paragraph of claim 4 (the preimage of $A\times A'$ being $\{\bar{\Sigma}^{\sharp,\mathsf{D}}_{s}\in A\}\cap\mathsf{D}^{-1}(A')$). Likewise the pair $(\Sigma_{s},W^{(s)})$ is measurable with respect to $\mathcal{F}$ and $\mathcal{P}(\mathbb{G}_{N})\otimes\mathcal{R}_{s}$: $\{\Sigma_{s}\in A\}=\bigcup_{y\in A}\bigcap_{\gamma}\{\Sigma^{\gamma}_{s}=y^{\gamma}\}\in\mathcal{F}$ for $A\subseteq\mathbb{G}_{N}$, each $\Sigma^{\gamma}_{s}$ being a random variable (claim 4), and $W^{(s)}$ is measurable by claim 1.

The recentred copy endpoint. Let $\mathbf{c}\in\mathbb{R}^{l}$ and define $\Xi:\mathbb{G}_{N}\times\mathbf{R}_{s}\to\mathbb{R}\times\mathbf{R}_{s}$ by $\Xi(x,r)=(\mathbf{c}\cdot\Psi_{s}(x,r),r)$. The first coordinate is measurable with respect to $\mathcal{P}(\mathbb{G}_{N})\otimes\mathcal{R}_{s}$ by claim 4 and claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and the second coordinate is the projection, measurable by claim 5 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable; by the rectangle argument, $\Xi$ is measurable with respect to $\mathcal{P}(\mathbb{G}_{N})\otimes\mathcal{R}_{s}$ and $\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$. Now $X''_{s}=\Psi_{s}\circ(\bar{\Sigma}^{\sharp,\mathsf{D}}_{s},\mathsf{D})$, so each component of $X''_{s}$ is the composition of the measurable pair with a measurable component of $\Psi_{s}$, hence a random variable on $(\Omega^{\sharp},\mathcal{F}^{\sharp},\mu^{\sharp})$ (a probability space by claim 4 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record); $|X''_{s}|\le2\sqrt{N}$ everywhere by the bound on $\Psi_{s}$ of claim 4; $\mathbf{c}\cdot X''_{s}$ is a random variable bounded by $2\sqrt{N}|\mathbf{c}|$ (The Cauchy-Schwarz Inequality on $\mathbb{R}^n$), hence square-integrable by the bounded-variable argument of claim 4; and $(\mathbf{c}\cdot X''_{s},\mathsf{D})=\Xi\circ(\bar{\Sigma}^{\sharp,\mathsf{D}}_{s},\mathsf{D})$ is measurable with respect to $\mathcal{F}^{\sharp}$ and $\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$ as a composition of measurable maps.

The law identity. Let $E\in\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$ and put $F'=\mathbf{1}_{\Xi^{-1}(E)}:\mathbb{G}_{N}\times\mathbf{R}_{s}\to[0,\infty]$, the indicator of the set $\Xi^{-1}(E)\in\mathcal{P}(\mathbb{G}_{N})\otimes\mathcal{R}_{s}$; $F'$ is measurable with respect to that $\sigma$-algebra (claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, the sets $\{F'>a\}$ being $\emptyset$, $\Xi^{-1}(E)$ or the whole space). Claim 3 of Law Identity between the N-Agent Aggregate Path with Its Observation Record and the Synthetic Copy's Regularised Path with Its Record, applied at the time $s$ of the horizon-$s$ instance, gives

$$\mathbb{E}\bigl[\mathbf{1}_{D_{0}}\,F'(\Sigma_{s},W^{(s)})\bigr]=P(D_{0})\int_{\Omega^{\sharp}}F'\bigl(\bar{\Sigma}^{\sharp,\mathsf{D}}_{s},\mathsf{D}\bigr)\,d\mu^{\sharp},$$

the record of the restricted solution being $W^{(s)}$. Here $F'(\bar{\Sigma}^{\sharp,\mathsf{D}}_{s},\mathsf{D})$ is the indicator of $\{(\mathbf{c}\cdot X''_{s},\mathsf{D})\in E\}$, so the right-hand side equals $\mu^{\sharp}(\{(\mathbf{c}\cdot X''_{s},\mathsf{D})\in E\})$ by The Integral of an Indicator Function is the Measure of the Set and $P(D_{0})=1$. On the left, $F'(\Sigma_{s},W^{(s)})$ is the indicator of $\{(\mathbf{c}\cdot\Psi_{s}(\Sigma_{s},W^{(s)}),W^{(s)})\in E\}$; by claim 4 this set differs from $\{(\mathbf{c}\cdot X'_{s},W^{(s)})\in E\}$ at most by points outside $\Omega_{0}$, and both sets belong to $\mathcal{F}$ (the first as the preimage of $\Xi^{-1}(E)$ under the measurable pair $(\Sigma_{s},W^{(s)})$, the second by claim 4). Now $P(\Omega_{0})=1$, the regular event of a solution having probability one by Solution of the Controlled N-Agent Dynamics, and $P(D_{0})=1$ by assumption, so $P(\Omega\setminus\Omega_{0})=0$ and $P(\Omega\setminus D_{0})=0$ by the additivity of the measure $P$. For events $A$, $A'$ that agree outside an event $\mathsf{N}$ with $P(\mathsf{N})=0$ one has $A\setminus\mathsf{N}=A'\setminus\mathsf{N}$ and, by additivity and monotonicity of $P$, $P(A)=P(A\setminus\mathsf{N})+P(A\cap\mathsf{N})=P(A\setminus\mathsf{N})$, since $0\le P(A\cap\mathsf{N})\le P(\mathsf{N})=0$; hence $P(A)=P(A\setminus\mathsf{N})=P(A'\setminus\mathsf{N})=P(A')$. Applying this with $\mathsf{N}=\Omega\setminus\Omega_{0}$ to the two events above, and noting that $\mathbb{E}[\mathbf{1}_{D_{0}}\mathbf{1}_{A}]=P(A\cap D_{0})$ by The Integral of an Indicator Function is the Measure of the Set and $P(A\cap D_{0})=P(A)$ by the same argument with $\mathsf{N}=\Omega\setminus D_{0}$, the left-hand side equals $P(\{(\mathbf{c}\cdot X'_{s},W^{(s)})\in E\})$. Hence

$$P\bigl(\{(\mathbf{c}\cdot X'_{s},W^{(s)})\in E\}\bigr)=\mu^{\sharp}\bigl(\{(\mathbf{c}\cdot X''_{s},\mathsf{D})\in E\}\bigr)\qquad\text{for every }E\in\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s},$$

which is the asserted equality of the two image measures (claim 1 of Image Measures, Measures with Densities, and Change of Variables).

The integral identity. Let $F:\mathbb{R}\times\mathbf{R}_{s}\to[0,\infty]$ be measurable with respect to $\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$, and write $\nu$ for the common image measure. By claim 2 of Image Measures, Measures with Densities, and Change of Variables applied to $P$ and the map $(\mathbf{c}\cdot X'_{s},W^{(s)})$, and then to $\mu^{\sharp}$ and the map $(\mathbf{c}\cdot X''_{s},\mathsf{D})$,

$$\mathbb{E}\bigl[F(\mathbf{c}\cdot X'_{s},W^{(s)})\bigr]=\int_{\mathbb{R}\times\mathbf{R}_{s}}F\,d\nu=\int_{\Omega^{\sharp}}F(\mathbf{c}\cdot X''_{s},\mathsf{D})\,d\mu^{\sharp}\qquad\text{in }[0,\infty].$$

This completes the proof.